Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6249 |
| automorphism | Aut(G) = #6249⋊C2 (order 432) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽3) ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | H(3,𝔽3) | C8 |
| Frattini, socle | C3 | AGL(1,𝔽9) |
| prime radical | #272 | C4 |
| cosocle | #1088 | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,3,5,10,6,11,16,12)(4,15)(7,13,19,8,21,14,26,23)(9,25,18,17,20,27,22,24);(1,18,22)(2,17,3)(4,26,19)(5,16,15)(6,11,24)(7,27,8)(9,10,14)(12,20,13)(21,23,25)Polycyclic presentation:
2^3,3^3|a^2=b,b^2=c,d^a=e,e^a=de^2,f^a=f^2,d^b=de^2,e^b=d^2e^2f^2,d^c=d^2f,e^c=e^2f^2,e^d=ef^213 conjugacy classes — view the full character table.
| maximal subgroup of | #6249⋊C2, C2×#6249, C3×#6249, … |
| maximal normal subgroup of | #6249⋊C2, C2×#6249, C3×#6249, … |
| maximal quotient of | #88586, C2×#6249, C3×#6249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 86);